CHAP, raj FEBMAT'S AND WILSON'S THEOREMS. 65 C. F. Gauss28 proved thafy if n is a prime, 2, 3, . . . , n 2 can be associated in pairs such that the product of the two of a pair is of the form xn+1. This step completes Schaffgotsch's25 proof of Wilson's theorem. Gauss29 proved Format's theorem by the method now known to be that used by Leibniz4 and mentioned the fact that the reputed proof by Leibniz had not then been published. Gauss30 proved that if a belongs to the exponent t modulo p, a prime, then a -a2 -a3. . .ae=( 1)(+1 (mod p). In fact, a primitive root p of p may be chosen so that a^p**"1^'. Thus the above product is congruent to p*, where Thus pft=(pV)'+1s(-l)<+1 (mod p). When a is a primitive root, o, a2, . . . , a""1 are congruent to 1, 2, . . . , p 1 hi some order. Hence (p 1) != ( l)p. This method of proving Wilson's theorem is essentially that of Euler.22 Gauss31 stated the generalization of Wilson's theorem: The product of the positive integers <A and prime to A is congruent modulo A to 1 if A =4, pm or 2pm, where p is an odd prime, but to +1 if A is not of one of these three forms. He remarked that a proof could be made by use of associated numbers28 with the difference that y?=l. (mod A) may now have roots other than =*= 1 ; also by use of indices and primitive roots30 of a composite modulus. S. F. Lacrobc32 reproduced Euler's13 third proof of Fermat's theorem without giving a reference. James Ivory33 obtained Format's theorem by a proof later rediscovered by Dirichlet.40 Let N be any integer not divisible by the prime p. When the multiples N, 2N, 3N, . . . , (p l)N are divided by p, there result p distinct positive remainders <p, so that these remainders are 1, 2,. . ., p 1 in some order.34 By multiplication, Np~lQ = Q+mp, where Q = (pl)\. Hence p divides Nv~l 1 since it does not divide Q. Gauss35 used the last method in his proof of the lemma (employed hi his third proof of the quadratic reciprocity law) : If k is not divisible by the odd prime p, and if exactly /x of the least positive residues of Jb, 2k, . . . , J (p - 1)& modulo p exceed p/2, then /b(p-1)/2= ( - iy (mod p). [Cf. Grunert.45] "Disquisitiones Arith., 1801, arts. 24, 77; Werke, 1, 1863, 19, 61. 29Disq. Arith., art. 51, footnote to art. 50. 30Disq. Arith., art. 75. 31Disq. Arith., art. 78. 32Complement des (Siemens d'algebre, Paris, ed. 3, 1804, 298-303; ed. 4, 1817, 313-7. 33New Series of the Math. Repository (ed. Th. Leybourn), vol. 1, pt. 2, 1806, 6-8. ğA fact known to Euler, Novi Comm. Acad. Petrop., 8, 1760-1, 75; Comm. Arith., 1, 275; and to Gauss, Disq. Arith., art. 23. Cf. G. Tarry, Nouv. Ann. Math., 18, 1899, 149, 292. 86Comm. soc. reg. sc. Gottingensis, 16, 1808; Werke, 2, 1-8. Gauss' Hohere Arith., German transl. by H. Maser, Berlin, 1889, p. 458.